High resolution numerical methods for compressible multi fluid flows and their applications in simulations

The developed methods aresubsequently applied to simulate Richtmyer-Meshkov instability RMI driven bycylindrical shock and shock-bubble interactions in two and three dimensions.Based on

COMPRESSIBLE MULTI-FLUID FLOWS AND THEIR APPLICATIONS IN SIMULATIONS

ZHENG JIANGUO

(B.S & M.E., University of Science and Technology of China,

Hefei, Anhui, P.R China)

A THESIS SUBMITTEDFOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2009

I would like to express my sincere gratitude to my supervisors, Associate Professors

T S Lee and S H Winoto for their advice, support and guidance during my thesisresearch

I am deeply grateful to my parents, my elder sisters as well as other members of

my family and my girl friend Hua Yi for their love and constant support Withoutthem, this work would have never been possible

I am indebted to Associate Professor Ma Dongjun at University of Science andTechnology of China for many good suggestions I also wish to thank Dr ZhangWeiqun at the Center for Cosmology and Particle Physics of New York Universityfor his help in implementation of adaptive mesh refinement

I would like to thank all my friends for their friendship and encouragement during

my four-year study at National University of Singapore

Finally, I am grateful to National University of Singapore for providing me with

a scholarship

Acknowledgments i

1.1 Background 1

1.2 Diffuse Interface Methods 5

1.2.1 Methods for Flows with Stiffened Gas EOS 5

1.2.2 Multi-fluid Flows with Mie-Gr¨uneisen EOS 7

1.2.3 Flows Involving Barotropic Components 10

1.3 Applications of Multi-fluid Algorithms in Simulations 11

1.3.1 Numerical Simulations of Richtmyer-Meshkov Instability 11

1.3.2 Shock-bubble Interactions 13

1.4 Objectives and Significance of Study 15

1.5 Outline of Thesis 16

ened Gas Equation of State 18

2.1 Governing Equations 19

2.1.1 Inviscid Model 19

2.1.2 Model with Viscous Effect and Gravity 22

2.2 Numerical Methods 24

2.2.1 Reconstruction of Variables 25

2.2.2 Unsplit PPM Scheme 32

2.2.3 Dimensional-splitting PPM Scheme 33

2.2.4 HLLC Riemann Solver 37

2.2.5 Two-shock Riemann Solver 38

2.2.6 Oscillation-free Property 41

2.2.7 Solution of the Diffusion Equations 43

2.3 Adaptive Mesh Refinement 44

2.4 Numerical Results 47

2.5 Summary 57

Chapter 3 Interface-capturing Methods for Flows with General Equa-tion of State 72 3.1 Governing Equations 73

3.2 Method Based on MUSCL-Hancock Scheme 77

3.2.1 Variables Reconstruction 77

3.2.2 HLLC Riemann Solver 79

3.2.3 Oscillation-free Property of the Present Method 82

3.3 Piecewise Parabolic Method 84

3.3.1 Unsplit PPM 84

3.3.2 Dimensional-splitting PPM 85

3.4 Adaptive Mesh Refinement 87

3.6 Numerical Results with PPM 96

3.7 Summary 97

Chapter 4 High-resolution Methods for Barotropic Two-fluid and Barotropic-nonbarotropic Two-fluid Flows 112 4.1 PPM for Barotropic-nonbarotropic Two-fluid Flows 113

4.1.1 Equation of State 113

4.1.2 Governing Equations 115

4.1.3 Lagrangian-remapping PPM for Multi-fluid Flows 117

4.1.4 Riemann Solver 121

4.1.5 Numerical Results 123

4.2 PPM for Barotropic Two-fluid Flows 127

4.2.1 Model Equations 127

4.2.2 Results of Numerical Simulations 129

4.3 Summary 130

Chapter 5 Numerical Simulations of Richtmyer-Meshkov Instability Driven by Imploding Shock 139 5.1 Richtmyer-Meshkov Instability Driven by Imploding Shock 139

5.1.1 Single-mode RMI 140

5.1.2 Effects of Shock Strength and Perturbation Amplitude on RMI 146 5.1.3 Random-mode Air-Helium Simulation 146

5.2 Summary 150

Chapter 6 Numerical Simulations of Shock-Bubble Interactions 157 6.1 Interactions of Shock with Helium Cylinder 157

6.1.1 Setup for Numerical Simulations 157

6.1.2 Results for M a = 1.2 159

6.2 Interactions of Shock with Helium Sphere 162

6.3 Interactions of Shock with Krypton Cylinder 166

6.4 Interactions of Shock with Krypton Sphere 168

6.5 Summary 170

This thesis is concerned with the development of high-resolution diffuse interfacemethods for resolving compressible multi-fluid flows The developed methods aresubsequently applied to simulate Richtmyer-Meshkov instability (RMI) driven bycylindrical shock and shock-bubble interactions in two and three dimensions.Based on ensemble averaging for multi-component flows, an inviscid compressiblemulti-fluid model is recovered The viscous effect and gravity can also be introducedinto the model The direct Eulerian piecewise parabolic method (PPM) is modifiedslightly and generalized to integrate numerically the hyperbolic part of governingequations Although the resulting dimensional-splitting and unsplit PPMs are com-plicated, they prove more accurate in interface capturing The present methodsare able to resolve the material interfaces sharply and deal with problems involvinghigh density and pressure ratios as well as large differences in equations of state(EOSs) across an interface The use of adaptive mesh refinement (AMR) allows us

to capture flow features at disparate scales

MUSCL-Hancock method is extended to resolve the multi-fluid flows with ponents modeled by Mie-Gr¨uneisen EOS which is referred to as general or complexEOS By adapting HLLC approximate Riemann solver to the advection equation,the volume fraction is updated properly As a result, the method proves very stableunder different situations, which is a remarkable advantage As Mie-Gr¨uneisen EOScan model a large number of real materials, this method can be applied to many

com-To simulate flows involving one or two barotropic components, methods based

on Lagrangian-remapping (LR) PPM are developed The basic idea is that themixtures of two fluids are considered to be nonbarotropic The solution procedure

is divided into Lagrangian step and separate remapping step The methods canproduce sharper profiles for discontinuities, particularly contact discontinuities thanother diffuse interface methods They prove quite stable and effective in dealing withthe multi-fluid flows

LR PPM is applied to numerically study RMI The results with our method forair-SF6 interface driven by a planar shock are found in good agreement with predic-tions of front tracking and theoretical models The evolution of single-mode air-SF6interface driven by an imploding shock is highly different from that of the planarcase The so-called reshock is observed In addition, random-mode perturbationsare imposed on air-helium interface to mimic real problems Random nature of theperturbations significantly alters evolution of the interface The effects of shockstrength and perturbation amplitude on RMI are also examined

The study also concentrates on the numerical investigation of cylindrical and

spherical bubbles in air accelerated by shock with Mach numbers (M a) in the range

of 1.2 ≤ M a ≤ 6 The bubbles may be lighter or heavier than the ambient air,

forming different configurations It is found that the time evolution of a specificbubble filled with helium or krypton is significantly altered by the shock strength.For three-dimensional (3D) bubbles, some new flow features observed in experimentsare reproduced numerically Our 3D results agree qualitatively well with the exper-imental images Some qualitative findings are reported on Mach number effects onthe bubbles evolutions

5.1 Air-SF6 simulation parameters 1515.2 Air-helium simulation parameters 1516.1 Properties of gases used in this study 172

2.1 Locations of characteristics for supersonic flow (a) and subsonic flow(b) The dashed lines labeled -, 0 and + correspond to characteristics

of speeds u − c, u and u + c, respectively Here, fluid is assumed to

propagate from left to right 592.2 The structure of a grid block In two dimensions, each block has

12 × 12 interior cells bounded by the dash-dot line and has 4 guard

cells at each boundary 592.3 The sketch of a simple computational domain covered with a set ofgrid blocks at different refinement levels The solid lines show outlines

of these blocks 602.4 The flux conservation at a jump in refinement The flux f on thecoarse cell interface should be equal to sum of the fluxes f1, f2 on thefine cell interfaces 602.5 Density contours of the square air bubble advection problem at time

t = 4 × 10 −4, using dimensional-splitting PPM The blue dashed linesshow the bubble outline at initial time 61

advection problem shown in Fig 2.5 Here, both the numerical cles) and exact (solid lines) solutions are shown The four subfiguresare (a) density, (b) pressure, (c) norm of the velocity vector and (d)volume fraction 622.7 Density contours of the circular air bubble advection problem at time

(cir-t = 4 × 10 −4, using unsplit PPM The blue dashed lines show thebubble outline at initial time 63

2.8 Cross-sectional results along the diagonal y = x for the air bubble

advection problem shown in Fig 2.7 64

2.9 Density contour of the 1D shock-interface interaction problem at t = 0.2, using dimensional-splitting PPM . 65

2.10 Cross-sectional solutions along the center line y = 0.5 to the problem

shown in Fig 2.9 Solid lines are the exact solution, while circles andsquares are numerical results from dimensional-splitting and unsplitPPMs, respectively 662.11 Schlieren images of the density and pressure fields for the underwa-ter explosion problem at 5 times The results are obtained usingdimensional-splitting PPM Here, ms denotes microsecond The out-lines of AMR blocks at end time are also illustrated 672.12 Results from unsplit PPM for problem in Fig 2.11 68

2.13 The density profiles along x = 0 for problem in Figs 2.11 and 2.12.

The solid lines denote Shyue (2006c)’s results; circles are solution

obtained using dimensional-splitting PPM with a 384 × 288 uniform

grid; diamonds and squares are results from dimensional-splitting andunsplit PPMs with AMR, respectively 69

The last two frames correspond to t = 2.0 . 69

2.15 A sequence of images of volume fraction contour at times 0, 50µs, 200µs, 350µs, 450µs, 600µs and 750µs The last two frames corre- spond to t = 750µs . 702.16 Iso-surfaces of volume fraction for the 3D Richtmyer-Meshkov insta-bility 71

2.17 The density and pressure profiles along x = 0.5 for 2D RMI at t = 2. 71

3.1 Log-Log plots of L1 errors vs grid size (a) and time step(b) 99

3.2 Cross-sectional results along y = 0.5 for a copper-explosive interface advection problem t = 240µs The dash-dot lines are solution with

AMR, while solid lines are solution with a uniform grid of 5000 cells.Frame (d) shows outlines of AMR blocks 100

3.3 Results of a copper-explosive impact problem at t = 85µs 101 3.4 Results of a detonation product-copper Riemann problem at t = 73µs.102

3.5 Results of the interaction between a shock in molybdenum and

molybdenum-MORB interface at t = 120µs 103 3.6 Results of a liquid-gas shock tube problem at t = 240µs 104

3.7 Schlieren images of density field for shock-helium bubble interaction 1053.8 Schlieren images of density and pressure fields for the underwaterexplosion problem 106

3.9 Density profiles along x = 0 for the problem in Fig 3.8 Squares

and solid lines are solutions obtained using the present and Shyue’smethods, respectively 1073.10 Schlieren images of density and pressure fields for shock interactionswith a block of MORB 108

using dimensional-splitting PPM The blue dashed line shows theplate outline at initial time 109

3.12 Cross-sectional results along the diagonal y = x for the copper plate

problem shown in Fig 3.11 Here, both the numerical (circles) andexact (solid lines) solutions are shown The four subfigures are (a)density, (b) pressure, (c) norm of the velocity vector and (d) volumefraction 1103.13 Schlieren images for the shock-bubble interaction problem 1113.14 Results for problem in Fig 3.13 from Nourgaliev et al (2006) 1114.1 Results of a shock interactions with a material interface The foursubfigures are (a) density(logarithm scale) , (b) pressure(logarithmscale), (c) momentum and (d) volume fraction of water The solidlines and circles denote the exact and computational solutions at time

t = 0.01, respectively. 1324.2 Density contours of 2D material interface advection problem at times

t = 0, 0.005 In this calculation, a 100 × 100 grid is used 132 4.3 Results of variables along diagonal y = x at time t = 0.005 for the

problem demonstrated in Fig 4.2 (a) density, (b) pressure, (c)velocity, (d) volume fraction of water Here, the velocity v is defined

as v = √ u2+ v2 The solid lines denote the exact solution, whilecircles give the numerical results obtained by PPM with a uniform

100 × 100 grid 133

4.4 Density and pressure contours of Richtmyer-Meshkov instability where

a liquid-gas interface is driven by a Mach 1.95 shock in the liquid The

numerical results are obtained by PPM with a 400 × 100 mesh grid. 134

used to study convergence of the numerical solution 1344.6 Cross-sectional plots of the density and pressure along the horizontal

center line y = 0.5 at time t = 2.0 The grids used are 100 × 25,

200 × 50 and 400 × 100 135

4.7 Contours of the density, pressure and volume fraction for the problem

of shock interactions with an air bubble Here a 200 × 240 mesh grid

is used 136

4.8 Density (left) and pressure (right) contours at time t = 0.003 taken

from Shyue (1999b) for comparison with results illustrated in Fig 4.7.137

4.9 Results for a two-fluid Riemann problem at t = 0.01 (a) density,

(b) pressure, (c) momentum, (d) volume fraction of the water-likematerial The solid lines represent the extract solution, while thecircles give the results by PPM with a uniform 200-cell grid 1374.10 2D results for the shock interactions with an air bubble Threecolumns (from left to right) are contours of density, pressure andvolume fraction of the water-like material, while two rows correspond

to two selected times t = 0.005, 0.01 In this calculation, a 400 × 240

mesh grid is used 138

4.11 Profiles of variables along the horizontal line y = 0 for the case of

2D shock-bubble interactions shown in Fig 4.10 The two rows

cor-respond to two times t = 0.005, 0.01, respectively Here, the density

profile of Shyue (2004), denoted as red dashed lines, is included forcomparison 1385.1 Initial configuration of RMI with a single-mode perturbation in cylin-drical geometry 1515.2 Evolution of interface for the planar air-SF6 simulation 151

unstable interface illustrated in Fig 5.2 For comparison, the predic-tions of the impulsive model, linear theory, front tracking and PPM

are all plotted 152

5.4 Density contour of the air-SF6 interface driven by an imploding shock at 1000µs 152

5.5 Amplitude vs time and growth rate vs time for the unstable air-SF6 interface in cylindrical geometry shown in Fig 5.4 153

5.6 Density contour of the air-helium interface driven by an imploding shock at 150µs 153

5.7 Amplitude vs time and growth rate vs time associated with the sim-ulation in Fig 5.6 154

5.8 Variations of amplitude and growth rate with time for the cylindri-cal air-helium simulation Here, shock of different Mach number is employed to examine effect of shock strength on RMI 154

5.9 Variations of amplitude and growth rate with time for the cylindri-cal air-helium simulation The amplitude of initial perturbation is changed to study its influence on RMI 155

5.10 Density contour of air-helium simulation with random-mode pertur-bation in cylindrical geometry 155

5.11 Amplitude vs time and growth rate vs time for problem in Fig 5.10 156 6.1 Schematic of the 2D computational domain 172

6.2 Schlieren images of density field for a helium cylinder, M a = 1.2 172

6.3 Contours of vorticity field for a helium cylinder, M a = 1.2 173

6.4 Helium cylinder, M a= 2 173

6.5 Helium cylinder, M a= 3 173

6.6 Helium cylinder, M a= 6 174

6.8 Vorticity field for a helium cylinder, M a= 3 174

6.9 Vorticity field for a helium cylinder, M a= 6 174

6.10 Density iso-surfaces for helium sphere, M a = 1.2 175

6.11 Density contours for helium sphere, M a = 1.2 176

6.12 Helium sphere, M a = 1.5 177

6.13 Experimental (left) and numerical (right) images for a helium sphere, M a = 1.5 The images are obtained from Layes and Le Metayer (2007).178 6.14 Volume fraction iso-surfaces for helium sphere, M a= 2 179

6.15 Helium sphere, M a= 3 180

6.16 Helium sphere, M a= 4 181

6.17 Helium sphere, M a= 5 182

6.18 Helium sphere, M a= 6 183

6.19 Krypton cylinder, M a = 1.2 184

6.20 Vorticity field for krypton cylinder, M a = 1.2 184

6.21 Krypton cylinder, M a= 2 184

6.22 Krypton cylinder, M a= 3 184

6.23 Krypton cylinder, M a= 6 185

6.24 Density iso-surfaces for krypton sphere, M a = 1.2 185

6.25 Density contours for krypton sphere, M a = 1.2 186

6.26 Volume fraction iso-surfaces for krypton sphere, M a= 2 187

6.27 Krypton sphere, M a= 3 188

6.28 Krypton sphere, M a= 4 189

6.29 Krypton sphere, M a= 5 190

6.30 Krypton sphere, M a= 6 191

Roman letters:

a amplitude of perturbation (chapter 5)

A Jacobian matrix

e internal energy per unit mass

E total energy per unit mass

E i error indicator

F inviscid numerical flux

Fa inviscid flux vector in the x direction

Fd viscous flux vector in the x direction

g gravitational acceleration

Ga inviscid flux vector in the y direction

Gd viscous flux vector in the y direction

Ha inviscid flux vector in the z direction

Hd viscous flux vector in the z direction

u x component of the velocity

U vector of primitive variables (chapter 2)

U vector of conserved variables

v velocity vector

W vector of primitive variables

x spatial position vector

Y (i) volume fraction of ith fluid

z (i) mass fraction of ith fluid

1.1 Background

Compressible multi-fluid flows, where immiscible fluids with fully different propertiesare separated by material interfaces, arise in many practical applications from iner-tial confinement fusion to combustion, underwater explosion, bubbly flow, etc Suchflow problems are also involved in astrophysics, and a typical example is Richtmyer-Meshkov instability (RMI), which plays an important role in supernova explosion.Since they are widespread, study on the multi-fluid flows has received much atten-tion and made significant progress in the computational fluid dynamics communityover the past few decades

Up to now, quite a few numerical methods have been developed to resolve thecompressible multi-fluid flows These methods can roughly be categorized into fourfamilies: front-tracking, volume of fluid, level-set and diffuse interface methods Inthis section, a brief review on developments of the first three types of methods will

be presented

The basic idea of front-tracking methods of Glimm et al (1981, 1998, 2000,2001a,b) is to introduce a lower dimensional grid which dynamically moves withvarious waves, such as shock, contact discontinuity, rarefaction and material inter-face The lower dimensional grid is coupled with a higher dimensional backgroundgrid Using Riemann problems solutions, a special algorithm is developed to ad-vance points on the lower dimensional grid In addition, see work of LeVeque and

Shyue (1996).

The front-tracking methods do not introduce numerical diffusions on ities, which distinguishes them from any interface-capturing method The methodstrack the discontinuities, implement correct jump conditions and thus keep all dis-continuities sharp As a large number of grid points are distributed on the interfaces,they are more accurate than other methods and small scale flow features on the in-terfaces can be resolved

discontinu-However, front tracking is very complicated when applied to deal with largeinterface deformations Another disadvantage is that the methods may becomeunstable when extremely small grid size or volume is present In addition, it isdifficult to extend the methods to three dimensions

Another family of numerical methods for the multi-fluid flows is volume of fluid(VOF) methods VOF methods define volume fractions of fluid components anduse them to reconstruct the material interfaces Suppose two fluids coexist in a

computational domain Ω In each grid cell, volume fraction of fluid 1, α, is defined

as ratio of volume of fluid 1 to that of the cell It is proved that the volume fractionobeys the following equation,

∂α

where v is velocity vector α lies in the interval [0, 1] In each cell, if value of α

is known, the interface can be reconstructed using various algorithms After theinterface location is determined, governing equations can be discretized and solved.The interface reconstruction is a key ingredient of VOF methods In two dimen-sions, the interface in a cell containing two fluids are considered to be made up ofcontinuous, piecewise smooth lines Many algorithms were developed to reconstructthe interface Hirt and Nichols (1981) represented the interface as line aligned withthe grid This method is simple but only first-order accurate To improve accuracy,

piecewise linear segments are constructed to approximate the interface, which leads

to the piecewise linear interface construction (PLIC) methods (Ashgriz and Poo,1991; Parker and Youngs, 1992; Rider and Kothe, 1995, 1998; Rudman, 1997, 1998;Pilliod Jr and Puckett, 1997, 2004; Gueyffier et al., 1999; Ma, 2002; Sethian andSmereka, 2003) The review article by Scardovelli and Zaleski (1999) is particularlyrecommended

VOF methods were extended to compressible multi-fluid flows by Parker andYoungs (1992), Miller and Puckett (1996) and Shyue (2006c)

One of advantages of VOF methods is that the material interfaces can be resolvedsharply as they are reconstructed accurately in each time step As the interfacesare defined implicitly, VOF can handle merger and breakup of the interfaces moreeasily than front tracking In addition, extending VOF from two dimensions to threedimensions is relatively simple

A principal disadvantage of VOF methods is the interface reconstruction rithms may be very complicated Besides, the geometric quantities such as curvaturecannot be computed in a straightforward manner Finally, the methods are complex

algo-in case of drastic topological changes of the algo-interfaces

Next, we look at level-set methods Osher and Sethian (1988) first proposedlevel-set methods, which are based on an implicit formulation of the interfaces.Consider a two-fluid problem in domain Ω = Ω1SΓ(t)SΩ2 Γ(t) is a material

interface separating fluid 1 in region Ω1 and fluid 2 in region Ω2 One is concerned

with motion of the interface Γ(t) To track the interface, Osher and Sethian (1988) constructed a function, φ(x, t), and set the zero-level set of this function to be the interface Γ(t) at any time, that is, Γ(t) = {x ∈ Ω : φ(x, t) = 0} Obviously, on the

can take different forms Usually, the value of φ(x, t) is set to be signed distance

between a given point x and the material interface

Typically, the level-set solution procedure consists of the following four steps:

1 Initialize the flow variables and level set function φ(x, t n)

2 Solve the level-set equation and update φ(x, t n) to new time level

3 Re-initialize φ(x, t n+1) so that value of level-set function is still the signeddistance to the interface

4 Solve the governing equations to update the flow variables and perform somespecial treatments on the interface

Mulder et al (1992) are the first to apply level-set methods to compute interfacemotion in compressible gas dynamics However, this method produced unphysicalpressure oscillations on the interfaces To eliminate the oscillations, Karni (1994,1996) employed the non-conservative form of the level-set equation so that the pres-sure is continuous across an interface Fedkiw et al (1999) proposed the ghost fluidmethod by defining the ghost cells on each side of the interface Although the ghostfluid method has some remarkable advantages, it cannot be applied to problemsinvolving a strong shock colliding with a material interface Liu et al (2003) modi-fied the original ghost fluid method Nourgaliev et al (2006) implemented level-setmethod in an adaptive mesh refinement environment

Level-set methods have some advantages First, the methods are easy to plement and can be extended to three dimensions quickly Second, the level-setmethods can deal with complex topological changes and calculate geometric quan-tities easily The methods also exhibit disadvantages They, for instance, are notdiscretely conservative, which can lead to some numerical errors For more infor-mation on level-set methods, the reader is referred to review article by Osher andFedkiw (2001) and that by Sethian and Smereka (2003)

im-1.2 Diffuse Interface Methods

Contrary to the preceding methodologies, diffuse interface methods do not trackthe material interfaces accurately, but allow them to diffuse numerically over afew grid cells Typically, model equations consist of Euler equations supplementedwith some governing equations for equation of state (EOS) parameters or passivelyadvected quantities, such as mass or volume fractions of components The materialinterfaces are identified using the EOS parameters or advected quantities However,

a problem is that there are numerical transition layers between different components.Obviously, one has to derive a mixture or artificial EOS to recover thermodynamicvariables such as pressure in the layers

As compared with three types of methods discussed in the previous section,diffuse interface methods are less accurate as they may admit excessive numericaldiffusions on the interfaces However, the methods are easier to implement andcan handle drastic topological changes As methods developed in this thesis alsobelong to the diffuse interface methods, in the following, we will discuss this type ofmethods in more detail

1.2.1 Methods for Flows with Stiffened Gas EOS

It is well known that the pressure oscillations occur on the material interfaces whenconventional finite difference or volume methods are applied to compressible multi-fluid flows calculations directly This is because that the methods cannot guaranteethat the pressure is continuous across a material interface

To eliminate the oscillations, Abgrall (1996) proposed a quasi-conservative proach for two-gas flows with ideal EOS The basic idea behind this method is thatfor a one-dimensional (1D) interface advection problem, where the pressure and ve-locity are in equilibrium throughout the domain and only the density is allowed tovary across the interface, the calculated pressure and velocity should remain equi-

ap-librium at any time Based on this condition and energy equation, Abgrall (1996)

derived a scheme for updating 1/(γ − 1), which corresponds to discretization of the advection equation for 1/(γ − 1) where γ is ratio of specific heat In fact, Ab-

grall (1996) gave up the conservative form of governing equations This method isoscillation-free and can handle strong shocks However, it is only applicable to 1Dproblems

Later, method of Abgrall (1996) was extended to multi-fluid flows modeled bystiffened gas EOS in multiple dimensions by Saurel and Abgrall (1999b) Approx-imate Riemann solvers including HLL and Roe schemes as well as exact Riemannsolver are used in conjunction with MUSCL scheme The method is robust, but due

to inherent diffusion of MUSCL scheme, the interfaces are smeared greatly Shyue

(1998) also generalized Abgrall’s idea to flows with stiffened gas EOS, deriving

γ-based and volume-fraction-γ-based models The non-conservative model equationsare solved using high-resolution wave propagation method In addition, Zheng et al.(2008a) developed method based on unstructured grid

Most of existing multi-fluid approaches are second order accurate To improveaccuracy, Johnsen and Colonius (2006) applied third- and fifth-order finite volumeweighted essentially non-oscillatory (WENO) schemes to interface capturing Theyare the first to adapt HLLC approximate Riemann solver to multi-fluid problems.Their method is high-order accurate and quasi-conservative However, as stated byShu (1997), the finite volume WENO scheme is expensive in two dimensions.Besides the above quasi-conservative methods, a fully conservative scheme withHLL Riemann solver was proposed by Wackers and Koren (2005) Marquina andMulet (2003) developed a flux-split algorithm based on conservative formulation oftwo-fluid model Nevertheless, it seems that this method cannot strictly eliminateoscillations when applied to calculations of shock-bubble interactions Bates et al.(2007) simulated a rectangular block of SF6 accelerated by shock using a multifluid

algorithm, which is conservative But this method is restricted to ideal gas EOS.

At the same time, there have been some attempts to derive models with severalvelocities and pressures, where the rate at which mechanical equilibrium betweendifferent phases is reached is taken into account, see Saurel and Abgrall (1999a) andMurrone and Guillard (2005)

Majority of the above methods ignore complex physical effects on the interfaces

To model multi-fluid flows with gravity, viscous effects and surface tension, Perigaudand Saurel (2005) developed a quasi-conservative formulation

Although the methods previously discussed are successful in capturing the terfaces, they suffer from some drawbacks First, due to the inherent numerical dif-fusions of these methods, excessive diffusions may be introduced into the interfaces,which is the principal disadvantage of the diffuse interface methods Sometimes, amaterial interface is spread over 10 cells Second, some of these methods are based

in-on a uniform Cartesian grid However, compressible flows usually involve variousflow features at disparate scales, such as steep gradients as well as smooth struc-tures These features should be resolved with different grid resolutions If a uniformgrid is used, the computational cost will be increased significantly when the grid isrefined Finally, most of these methods ignore physical effects such as viscosity.Therefore, it is desirable to develop simple and robust numerical methods whichare less diffusive than those previously mentioned In addition, we wish to employeadaptive mesh refinement (AMR) to improve grid resolution in local regions con-taining sharp structures such as shocks and material interfaces In some problems,viscosity plays an important role and so it should be taken into account

1.2.2 Multi-fluid Flows with Mie-Gr¨uneisen EOS

Although much research has been devoted to resolution of multi-fluid flows, fewstudies have been done on fluids modeled by general or complex EOS In this thesis,

the general or complex EOS refers to Mie-Gr¨uneisen EOS, which includes cases

of stiffened gas, van der Waals, Jones-Wilkins-Lee EOSs, etc These EOSs canmodel a variety of real materials in practice, but they also introduce difficulties intonumerical algorithms

Miller and Puckett (1996) developed a conservative Godunov-type algorithmfor computing multiphase flow problems with each of the phases modeled by Mie-Gr¨uneisen EOS In this algorithm, the multiphase mixtures are modeled as an effec-tive phase with single-valued pressure and velocity which is a fundamental assump-tion The resulting model system is hyperbolic and solved using high-resolutionGodunov-type method

This algorithm can be applied to problems involving materials in condensedphases such as liquids and solids under high pressure for example It is based onvolume of fluid formulation, which requires one to reconstruct the material inter-faces at each time step to determine individual phase volumes advected across cellinterfaces As we know, the interface reconstruction procedure is not trivial Inaddition, this algorithm uses a simplified multiphase model One has to relax thepressure to reach mechanical equilibrium in cells containing more than one material.Saurel and Abgrall (1999a) also proposed a numerical model for compressibletwo-phase flows In this model, pressure and velocity relaxations between two phasesare considered The resulting model equations are complex, consisting of mass,momentum and energy equations for each phase as well as an advection equationfor volume fraction This method seems robust

However, non-conservative terms accounting for source and relaxation terms inthe equations lead to difficulties in solving the model system The authors calculatednumerical fluxes using modified Godunov-Rusanov scheme and approximate HLLRiemann sovler, and the resulting algorithms are diffusive

By using the assumption of pressure and velocity equilibriums across a

mate-rial interface, Shyue (1999a, 2001) also developed high-resolution wave propagationmethods for flows with van der Waals and Mie-Gr¨uneisen EOSs The algorithmscombine Euler equations with a set of transport equations for material-dependentquantities in mixture EOS These quantities are used to calculate the pressure ofmixtures.

One disadvantage of Shyue’s methods is that the transport equations depend onthe specific form of EOS For general EOS, the model equations may become verycomplex

Later, a simpler five-equation model was introduced by Allaire et al (2002)

In one dimension, the governing equations consist of mass conservation equationfor each fluid, momentum and energy equations for the mixtures and an advectionequation for volume fraction of one fluid component All the material-dependentparameters of the mixture EOS can be calculated from partial densities and volumefraction The assumption of pressure equilibrium or isobaric closure is employed toclose the model system This model is very simple and does not vary with the form

of EOS

In this method, an advection scheme is developed to update the volume fraction.But it may be unstable under certain situations Physically, the volume fraction ispassively advected at local speed Nevertheless, some of methods in the literatureupdate value of the volume fraction according to the velocity field at previous timestep This, however, can lead to unphysical partial densities and failure of calcula-tions in a number of situations

As discussed above, it is necessary to develop effective algorithms to model fluid flows with general EOS In this thesis, we construct high-resolution methodswith AMR, which are very robust and stable

multi-1.2.3 Flows Involving Barotropic Components

Practical applications often involve barotropic fluids, that is, fluids whose pressureonly depends on density A typical example is water modeled by Tait EOS Forbarotropic flows, the energy equation dose not need to be solved as the pressure can

be calculated from the density directly

While many studies have been done on nonbarotropic multi-fluid flows, littleliterature is available on flows with two barotropic fluids van Brummelen and Ko-ren (2003) developed a non-oscillatory conservative method for barotropic two-fluidflows with Tait EOS Besides, Shyue (2004) presented an interface-capturing method

to handle such flow problems The basic idea of Shyue (2004) is to derive a isentropic form of Tait EOS to model the mixtures of two barotropic components

non-In transition layers between the two fluids, the energy equation is coupled to theisentropic Euler equations However, in regions containing a pure barotropic fluidonly, the Euler equations in isentropic form are employed

Although algorithm of van Brummelen and Koren (2003) was shown to be valid,

it was only applied to one-dimensional cases Shyue (2004)’s method can resolveproblems in multiple dimensions However, with this method, the material interfacesdiffuse greatly

On the other hand, there are a large number of two-fluid problems involving

a barotropic component and a nonbarotropic one Shyue (2006b) generalized hismethod in Shyue (2004) to barotropic and nonbarotropic two-fluid problems Again,the mixtures of two components are considered to be nonbarotropic and a mixtureEOS is derived based on the assumption of pressure equilibrium

In this method, the nonbarotropic fluid component is modeled by Noble-AbelEOS, which is mainly used to describe gas Therefore, it is desirable to use amore complex EOS such as stiffened gas EOS to model more materials In ad-dition, a big challenge to Shyue (2006b)’s method is that energy conservation for

the nonbarotropic component cannot be maintained in small neighborhoods of theinterfaces This issue is still open.

As discussed above, it is necessary to develop numerical methods to deal withbarotropic two-fluid and barotropic-nonbarotropic two-fluid flows The methodsshould produce the sharp material interfaces and can handle flows with more com-plex EOS

1.3 Applications of Multi-fluid Algorithms in

Simula-tions

1.3.1 Numerical Simulations of Richtmyer-Meshkov Instability

When an incident shock collides with a material interface separating two differentfluids, the interface becomes unstable and evolves with time This instability wasfirst predicted theoretically by Richtmyer (1960) and then studied experimentally

by Meshkov (1970) Since then, it is referred to as Richtmyer-Meshkov instability(RMI) This instability plays important roles in some astrophysical phenomena andengineering applications, and so has been studied extensively Brouillette (2002)presented a detailed review on the theoretical, experimental as well as numerical de-velopments of RMI during past 40 years In addition, see review article by Zabusky(1999) for more information Here, we only provide a brief review of RMI driven bycylindrical shocks

Zhang and Graham (1998) carried out a detailed numerical investigation of RMItriggered by imploding and exploding shocks, which refer to shocks moving radiallyinwards and outwards, respectively, in a cylindrical coordinate system The shockscan collide with the material interfaces from heavy fluid to light fluid or from lightone to heavy one Therefore, four configurations are examined In that study, thephenomena including reshock, phase inversion and twice phase inversion, effects of

number of fingers and initial perturbation amplitude, and growth rates ing to spikes and bubbles are all studied carefully A significant difference betweenplanar and cylindrical RMIs is the reshock For example, in configuration where animploding shock accelerates an interface from light phase to heavy one, the so-calledreshock leads to phase inversion of the material interface.

correspond-Later, Saltz et al (1999) investigated RMI driven by imploding shocks usingtwo Eulerian hydrodynamics codes Single-mode and multi-mode perturbations areimposed on the interfaces Some preliminary results are presented

Although study of Zhang and Graham (1998) provides an insight into mechanism

of cylindrical RMI, they only investigated single-mode perturbations However,there are almost no regular perturbations in real problems So it is necessary toconsider random-mode perturbations to mimic real problems It is believed thatrandom initial amplitude and wavelength significantly complicate interface evolutionprogress, and there are interesting phenomena peculiar to this problem

On the other hand, Saltz et al (1999) did not investigate the case where ploding shocks are located in heavy gas initially

im-Here, we use high-resolution piecewise parabolic method (PPM) to simulate RMI

in a cylindrical geometry Our simulations are motivated by experiment of Benjamin

et al (1993) and that of Meshkov (1970) Numerical results in cylindrical try are compared with those in planar geometry to show differences In addition,random-mode case is also studied

geome-Besides the above cylindrical RMI, in recent years, there have been some studieshandling RMI for axisymmetric flows in spherical domains, for example, see Dutta

et al (2004) and Glimm et al (2002)

1.3.2 Shock-bubble Interactions

Shock-bubble interactions induce RMI since on the bubble interface, the density andpressure gradients are not colinear after shock passage It is well known that RMIoccurs in the broad area of applications Therefore, bubble accelerated by shockhas received much attention during the past few decades It should be noted that

in this thesis, bubble refers to cylinder in two dimensions and to sphere in threedimensions

Up to now, a single bubble driven by an incident shock has been extensivelystudied experimentally In the pioneering work of Haas and Sturtevant (1987) ,time evolutions of helium and refrigerant 22 (R22) gas cylinders and spheres werepresented using shadowgraph pictures Jacobs (1993) and Zoldi (2002) also con-ducted experiments to record cylinder deformations under weak shock by means ofplanar laser-induced fluorescence (PLIF) Besides the above single bubble configura-tion, Tomkins et al (2002, 2003) and Kumar et al (2005, 2007) studied interactions

of two or three cylinders with weak shock More recently, Layes et al (2003, 2005)carried out experiments to investigate evolution of gas sphere accelerated by weakshock The gas, helium or nitrogen or krypton, is filled in the sphere in air at at-mospheric pressure to produce different density jumps across the sphere interface

In addition to weak shock, Ranjan et al (2005, 2007) studied soap-film sphere filledwith argon or helium impacted by strong shock The surrounding gas is atmosphericnitrogen

On the other hand, the shock-bubble interactions have been investigated ically Quirk and Karni (1996) reproduced the experiments of Haas and Sturtevant(1987), providing a comprehensive view of bubbles deformations For example, inthe case of a shock interacting with a heavy R22 cylinder, the refracted shock is wellidentified and it is found, for the first time, that the shock thickening results fromgeneration of a compression system matching the pressure jumps between the weak

numer-and strong parts of the refracted shock Besides, Quirk numer-and Karni (1996) proposed

a method for producing Schlieren images of density and pressure fields to visualizeweak waves in the solution and to compare numerical and experimental results

In the above work, a sharp interface is assumed In contrast, Zoldi (2002) ulated evolution of a diffusive SF6 cylinder as in his experiment, one gas is injectedinto a shock tube filled with the other gas to generate a gas column Therefore, a dif-fusive interface exists The numerical results are in good qualitative agreement withexperimental images However, there are some quantitative differences between thecalculated and measured length measurements of the cylinder, which is attributed

sim-to the fact that one cannot acquire accurate initial conditions from the tal image at initial time Zhang et al (2004) optimized the initial conditions andachieved better agreement with the experimental data of Zoldi (2002) In addition

experimen-to these works, the shock-bubble interactions were also simulated in other studies,for example, see Palekar et al (2000), Marquina and Mulet (2003), Shyue (2006c),Johnsen and Colonius (2006) and Nourgaliev et al (2006)

Although many efforts have been made to resolve shock-accelerated bubble, inmost of the previous studies, only weak shock is considered With strong shock,the behaviors of bubble may change significantly and some new flow features mayappear Hence, it is necessary to simulate bubble driven by strong shock Bagabir

and Drikakis (2001) investigated the Mach number (M a) effects on the evolution of

a helium cylinder by considering M a in the range of 1.22 ≤ M a ≤ 6 However, they

set ratios of specific heat of both air and helium to be 1.4 to use a single-componenthydrodynamics code to resolve two-gas flows where gas components have differentthermodynamic properties This, however, may affect the cylinder evolution So, it

is desired to employ a two-fluid model In addition, heavy density cylinder shouldalso be taken into account as its evolution process is completely different from that

of the helium cylinder

So far, relatively little work has been done on numerical investigation of accelerated sphere in three dimensions Giordano and Burtschell (2006) simulatedexperiment of Layes et al (2003) by solving two-dimensional (2D) axisymmetricflow model equations They examined differences in behaviors between cylinder andsphere driven by a Mach 1.2 shock Layes and Le Metayer (2007) compared nu-merical results with experimental images and achieved good qualitative agreement.Again, the 2D axisymmetric model is used More recently, Niederhaus et al (2008)and Ranjan et al (2008) carried out 3D numerical simulations of shock-sphere in-teractions.

shock-However, in most of the above studies, only weak shock is considered There is

no a comprehensive study of Mach number effects on shock-sphere interactions inboth convergent and divergent configurations Hence, it is necessary to carry outsimulations to examine Mach number effects on sphere evolution

1.4 Objectives and Significance of Study

The purpose of this study is to develop high-resolution numerical methods for ing compressible multi-fluid flows with various EOSs In addition, the methods areapplied to simulate RMI driven by cylindrical shocks and shock-bubble interactions.The more specific aims are as follows:

resolv-1 To develop direct Eulerian PPM algorithm for multi-fluid flows with ened gas EOS The inviscid multi-fluid model is recovered using theory of multi-component flows Viscous effects and gravity can also be taken into account AMR

stiff-is employed to resolve various flow features at dstiff-isparate scales

2 To develop numerical methods based on MUSCL and PPM schemes to late multi-fluid flows where fluid components are modeled by Mie-Gr¨uneisen EOS.The methods should be stable and robust under different situations

simu-3 To develop Lagrangian-remapping version of PPM to resolve barotropic

two-fluid and barotropic-nonbarotropic two-two-fluid problems.

4 To simulate RMI driven by imploding shocks Besides the single-mode turbations, the random-mode ones are imposed on the material interfaces to mimicreal problems

per-5 To simulate time evolutions of gas cylinders and spheres accelerated by shocksand examine Mach number effects on the evolution processes

Theoretically, the numerical methods developed in this study are accurate andefficient and therefore can be applied to resolve a wide range of real problems How-ever, the compressible multi-fluid flows are actually very complex It is impossible

to consider all factors in simulations Some assumptions are made in this study:

1 Our methods are based on the multi-fluid models with single pressure andvelocity In cells containing more than one fluid component, all phases are in pressureand velocity equilibriums In fact, the models have been used widely and proveeffective

2 In simulations of RMI and shock-bubble interactions, fluid components areassumed to be inviscid In addition, the flows may become turbulent at late times.However, turbulence model is not used in the present study

One contribution of this work is to apply high-resolution PPMs together withAMR to compressible multi-fluid flows modeled by different EOSs Another contri-bution is to extend HLLC Riemann solver to handle general EOS, which producesvery stable numerical methods Besides, some interesting physical phenomena areobserved in RMI driven by imploding shock and in shock-bubble interactions

1.5 Outline of Thesis

The thesis is organized into seven chapters:

In chapter 1, introduction and literature review are presented

In chapter 2, using the theory of multi-component flows, an inviscid compressible

multi-fluid model is recovered This model can be extended to include viscouseffects and gravity The dimensional-splitting and unsplit PPMs are generalized tointegrate the hyperbolic part of governing equations AMR is used to improve gridresolution in local regions.

In chapter 3, a MUSCL-type method with AMR is developed to resolve fluid flows with general EOS, that is, Mie-Gr¨uneisen EOS The HLLC approximateRiemann solver is generalized to solve Riemann problem and adapted to the advec-tion equation The reason some schemes for the advection equation are unstable isgiven The temporal and spatial accuracies of the method are validated BesidesMUSCL, PPM is also implemented

multi-In chapter 4, Lagrangian-remapping PPM is generalized to barotropic two-fluidand barotropic-nonbarotropic two-fluid flows Non-isentropic artificial EOSs arederived to model the two-fluid mixtures The methods are validated using a set oftest problems

In chapter 5, RMI with single- and random-mode perturbations driven by ploding shocks is investigated Results in planar and cylindrical geometries arecompared Effects of perturbation amplitude and shock strength are also studied

im-In chapter 6, shock-bubble interactions are simulated The cylindrical and ical bubbles are filled with helium or krypton The Mach number in the range of

spher-1.2 ≤ M a ≤ 6 is studied to investigate effects of shock strength on bubbles

evolu-tions The 3D results are compared with the experimental data and the differencesbetween 2D and 3D cases are presented

In chapter 7, conclusion of this study and recommendation for future work aregiven

Flows with Stiffened Gas Equation of State

In this chapter1, high-resolution methods are developed to capture the materialinterfaces of compressible multi-fluid flows in multiple dimensions Using the the-ory of multi-component flows, a fluid mixture model system with single velocityand pressure is recovered to model the current multi-fluid flows, and viscous effectand gravity are also considered in this system A consistent thermodynamic lawbased on the assumption of pressure equilibrium is employed to describe the ther-modynamic behaviors of the pure fluids and mixture of different components Thedimensional-splitting and unsplit piecewise parabolic methods (PPM) are extended

to numerically integrate the hyperbolic part of the model system, while the tem of diffusion equations is solved using an explicit, central difference scheme Theblock-structured adaptive mesh refinement (AMR) capability is built into the hydro-dynamic code to locally improve grid resolution The current methods are validatedusing a set of test problems and numerical results show their good performance

sys-It appears that the methods have some advantages over others First, the modelsystem is very simple with the viscous effect and gravity included Second, the meth-ods are capable of producing sharper representation of discontinuities, particularlycontact discontinuities Thus, they are highly suitable for the material interfacecapturing Third, the use of AMR allows different flow features at disparate scales

1 Part of this work has been published as:

Zheng, J G., Lee, T S., and Ma, D J (2007) A piecewise parabolic method for barotropic

two-fluid flows International Journal of Modern Physics C, 18(3):375–390 (Zheng et al., 2007).

to be resolved sufficiently Finally, the methods are quite robust.

In this section, a volume-fraction-based fluid mixture model is recovered first forinviscid, compressible multi-fluid flows Then, viscosity and gravity are added tothe inviscid model

2.1.1 Inviscid Model

Based on ensemble averaging, Drew and Passman (1999) derived equations of motionfor multi-component materials flows, where complex interactions between differentphases are considered This methodology can be adapted to the current flows torecover the governing equations

Suppose V is a control volume containing N components A component indicator

function is defined as,

with v0 being the velocity vector of a material interface

The volume average ¯g of a physical quantity g is defined to be,

¯

g = 1V

where S i is the material interface inside V separating component k and other fluids,

and nk is the unit external normal to component k For each component k governed

by the Euler equations, multiplying mass, momentum and energy equations by X k,respectively, and performing volume average for the resulting expressions and forthe topological equation (2.2) give (Drew and Passman, 1999; Shyue, 2006a),

with D = (v (k) − v0) · ∇X k In above equations, for component k, ρ (k) is density,

v(k) is velocity, p (k) is pressure and E (k) = e (k)+1

2v(k) · v (k)is the total energy with

e (k) being internal energy

As in Drew and Passman (1999), we present the averaged equations Here, it is

assumed that all components share the same pressure and velocity, that is, p (k) = p

and v(k) = v It is also required that the velocity of component k is equal to that

of the interface, that is, v(k) = v0 These crucial simplifications are made in such

a way that the various interfacial source terms are eliminated (Drew and Passman,

1999) The average of X k is,

Y (k)= 1

Z

Y (k) is known as volume fraction, that is, the ratio of volume of component k to

that of control region Then, the topological equation in (2.5) becomes an advection

equation for volume fraction Y (k),

where N is the number of components, and N = 2 for the two-fluid flows In this

system, the density, momentum and total energy are defined as,

The challenge is how to find a consistent thermodynamic law to close the system(2.9) so that the pressure oscillations do not appear on the material interfaces Inthis chapter, both components fulfill the stiffened gas EOS,

where p, ρ, e are the pressure, density and internal energy, respectively γ and π

are two material-dependent parameters This EOS is applicable to a wide range ofmaterials including gas, liquid and solid under high pressure

It is assumed that the artificial EOS for the mixture of two components takes thesame form as EOS (2.11) Then, according to relation (2.10b), the internal energy

of the mixture is given by,

ρe = p + γπ

γ − 1 =

2X

k=1

Y (k)ˆ(k)ˆ(k)=

2X

As mentioned earlier, both components should have the same pressure, p = p(1) =

p(2) Then, the above equation is split into two relations,

1

γ − 1=

2X

k=1

Y (k) γ

(k) π (k)

The solution for γ and π is found by solving the two equations Here, a key issue is

to employ the iso-pressure closure to eliminate the pressure oscillations, as discussed

in Allaire et al (2002) At this point, the inviscid model system is closed completely

2.1.2 Model with Viscous Effect and Gravity

Perigaud and Saurel (2005) proposed a quasi-conservative formulation for ible multi-fluid flows with viscous effect using asymptotic analysis and some approx-imations The system (2.9) is a simplified version of model in Perigaud and Saurel